Preheat the oven to 425 degrees F.
Line a baking sheet with aluminum foil.
In a medium pot over medium-high heat, add the chicken, the carrots, the celery, the onions, and the chicken broth and bring it to a simmer.
Simmer the chicken mixture, while stirring occasionally, until the chicken reaches an internal temperature of 165 degrees F and the vegetables are just fork tender, about 15 minutes.
Reserve 1 cup of the chicken broth, draining and discarding the remainder.
In a medium saucepan over medium heat, melt the butter.
Add 2 tablespoons of the flour to the melted butter and cook, while whisking, until combined and the flour is golden, about 30 seconds-1 minute.
Slowly add the reserved broth and 1/3 cup of the milk to the butter mixture, while whisking, until it is smooth.
Cook the gravy, while stirring constantly, until it thickens and begins to bubble, about 2-5 minutes.
Transfer the gravy from the heat.
Add the salt, the black pepper, and the chicken mixture to the gravy and stir to combine.
Divide the filling mixture between two 12-ounce ramekins.
In a small bowl, add the egg and the remaining milk and whisk to combine.
Using the extra flour, dust a work surface.
Add the pie crust to the prepared work surface and divide it evenly in two.
Roll each of the pie crust halves out so they are both 1/8-inch-thick.
Drape each of the pie crust halves over 1 of the ramekins and trim the edges so each has about 3/4-inch overhang.
Gently press the edges of the pie doughs in so the dough drapes over the sides of the ramekin slightly.
Using a sharp-tipped knife, cut a few slits in the top of each of the pot pies for steam to escape.
Brush the egg mixture over the top of each of the pot pies.
Place the ramekins on the prepared baking tray and bake until the tops are golden and the filling is bubbly, about 15-20 minutes.
Transfer the pot pies from the oven and let them cool for 5 minutes.
Serve.